Construction of the fermionic vacuum and of fermionic operators of creation and annihilation in the theory of algebraic spinors

TL;DR

This paper develops a mathematical framework for fermionic variables within Clifford algebras, constructing vacuums and operators that deepen the algebraic understanding of fermions beyond traditional spinor representations.

Contribution

It introduces fermionic variables as fundamental objects in Clifford modules, constructs primitive idempotents, and demonstrates their role in decomposing modules into minimal ideals.

Findings

Fermionic variables are more fundamental than spinors.

Primitive idempotents form a set of Clifford vacuums.

Modules decompose into direct sums of minimal ideals.

Abstract

We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.